Difference between revisions of "Euler problems/41 to 50"

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== [http://projecteuler.net/index.php?section=problems&id=41 Problem 41] ==
+
Do them on your own!
What is the largest n-digit pandigital prime that exists?
 
 
 
Solution:
 
<haskell>
 
import Data.List
 
isprime a = isprimehelper a primes
 
isprimehelper a (p:ps)
 
    | a == 1 = False
 
    | p*p > a = True
 
    | a `mod` p == 0 = False
 
    | otherwise = isprimehelper a ps
 
primes = 2 : filter isprime [3,5..]
 
problem_41 =
 
    head.filter isprime.filter fun $ [7654321,7654320..]
 
    where
 
    fun =(=="1234567").sort.show
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=42 Problem 42] ==
 
How many triangle words can you make using the list of common English words?
 
 
 
Solution:
 
<haskell>
 
import Data.Char
 
trilist = takeWhile (<300) (scanl1 (+) [1..])
 
wordscore xs = sum $ map (subtract 64 . ord) xs
 
problem_42 megalist=
 
    length [ wordscore a |
 
    a <- megalist,
 
    elem (wordscore a) trilist
 
    ]
 
main=do
 
    f<-readFile "words.txt"
 
    let words=read $"["++f++"]"
 
    print $problem_42 words
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=43 Problem 43] ==
 
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
 
 
 
Solution:
 
<haskell>
 
import Data.List
 
l2n :: (Integral a) => [a] -> a
 
l2n = foldl' (\a b -> 10*a+b) 0
 
 
swap (a,b) = (b,a)
 
 
explode :: (Integral a) => a -> [a]
 
explode =
 
    unfoldr (\a -> if a==0 then Nothing else Just $ swap $ quotRem a 10)
 
problem_43 = sum . map l2n . map (\s -> head ([0..9] \\ s):s)
 
                . filter (elem 0) . genSeq [] $ [17,13,11,7,5,3,2]
 
 
 
mults mi ma n = takeWhile (< ma) $ dropWhile (<mi) $ iterate (+n) n
 
 
sequ xs ys = tail xs == init ys
 
 
addZ n xs = replicate (n - length xs) 0 ++ xs
 
 
genSeq [] (x:xs) = genSeq
 
                  (filter (not . doub)
 
                  $ map (addZ 3 . reverse . explode) $ mults 9 1000 x)
 
                  xs
 
genSeq ys (x:xs) =
 
    genSeq (do
 
            m <- mults 9 1000 x
 
            let s = addZ 3 . reverse . explode $ m
 
            y <- filter (sequ s . take 3) $ filter (not . elem (head s)) ys
 
            return (head s:y)
 
          ) xs
 
genSeq ys [] = ys
 
 
 
doub xs = nub xs /= xs
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=44 Problem 44] ==
 
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
 
 
 
Solution:
 
<haskell>
 
import Data.Set
 
problem_44 =
 
    head solutions
 
    where
 
    solutions =
 
        [a-b |
 
        a <- penta,
 
        b <- takeWhile (<a) penta,
 
        isPenta (a-b),
 
        isPenta (b+a)
 
        ]
 
    isPenta = (`member` fromList  penta)
 
    penta = [(n * (3*n-1)) `div` 2 | n <- [1..5000]]
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=45 Problem 45] ==
 
After 40755, what is the next triangle number that is also pentagonal and hexagonal?
 
 
 
Solution:
 
<haskell>
 
isPent n =
 
    (af == 0) && ai `mod` 6 == 5
 
    where
 
    (ai, af) = properFraction $ sqrt $ 1 + 24 * (fromInteger n)
 
 
problem_45 = head [x | x <- scanl (+) 1 [5,9..], x > 40755, isPent x]
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=46 Problem 46] ==
 
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
 
 
 
Solution:
 
 
 
This solution is inspired by exercise 3.70 in ''Structure and Interpretation of Computer Programs'', (2nd ed.).
 
 
 
millerRabinPrimality on the [[Prime_numbers]] page
 
 
 
<haskell>
 
import Data.List
 
isPrime x
 
    |x==3=True
 
    |otherwise=millerRabinPrimality x 2
 
problem_46 =
 
    find (\x -> not (isPrime x) && check x) [3,5..]
 
    where
 
    check x =
 
        not $ any isPrime $takeWhile (>0) $ map (\y -> x - 2 * y * y) [1..]
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=47 Problem 47] ==
 
Find the first four consecutive integers to have four distinct primes factors.
 
 
 
Solution:
 
<haskell>
 
import Data.List
 
problem_47 = find (all ((==4).snd)) . map (take 4) . tails
 
                . zip [1..] . map (length . factors) $ [1..]
 
fstfac x = [(head a ,length a)|a<-group$primeFactors x]
 
fac [(x,y)]=[x^a|a<-[0..y]]
 
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
 
factors x=fac$fstfac x
 
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
 
 
 
primeFactors n = factor n primes
 
    where
 
        factor _ [] = []
 
        factor m (p:ps) | p*p > m        = [m]
 
                        | m `mod` p == 0 = p : [m `div` p]
 
                        | otherwise      = factor m ps
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=48 Problem 48] ==
 
Find the last ten digits of 1<sup>1</sup> + 2<sup>2</sup> + ... + 1000<sup>1000</sup>.
 
 
 
Solution:
 
If the problem were more computationally intensive, [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponentiation] might be appropriate.  With this problem size the naive approach is sufficient.
 
 
 
powMod  on the [[Prime_numbers]] page
 
 
 
<haskell>
 
problem_48 = flip mod limit$sum [powMod limit n n | n <- [1..1000]]
 
    where
 
    limit=10^10
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=49 Problem 49] ==
 
Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
 
 
 
Solution:
 
millerRabinPrimality on the [[Prime_numbers]] page
 
 
 
<haskell>
 
import Control.Monad
 
import Data.List
 
isPrime x
 
    |x==3=True
 
    |otherwise=millerRabinPrimality x 2
 
 
primes4 = takeWhile (<10000) $ dropWhile (<1000) primes
 
 
 
problem_49 = do
 
    a <- primes4
 
    b <- dropWhile (<= a) primes4
 
    guard ((sort $ show a) == (sort $ show b))
 
    let c = 2 * b - a
 
    guard (c < 10000)
 
    guard ((sort $ show a) == (sort $ show c))
 
    guard $ isPrime c
 
    return (a, b, c)
 
 
primes = 2 : filter (\x -> isPrime x ) [3..]
 
</haskell>
 
 
 
== [http://projecteuler.net/index.php?section=problems&id=50 Problem 50] ==
 
Which prime, below one-million, can be written as the sum of the most consecutive primes?
 
 
 
Solution:
 
(prime and isPrime not included)
 
 
 
<haskell>
 
import Control.Monad
 
findPrimeSum ps
 
    | isPrime sumps = Just sumps
 
    | otherwise    = findPrimeSum (tail ps) `mplus` findPrimeSum (init ps)
 
    where
 
    sumps = sum ps
 
 
 
problem_50 = findPrimeSum $ take 546 primes
 
</haskell>
 

Revision as of 21:46, 29 January 2008

Do them on your own!